Tagged Questions

2
votes
0answers
34 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
1
vote
4answers
171 views

What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 ...
2
votes
1answer
38 views

Bounding a continuously differentiable function using Taylor given the function is bounded by the norm of x

Suppose $0 < r < 1$ and that $f \colon B_1(0) \to \mathbb R$ is continuously differentiable. If there is an $\alpha > 0$ s.t. $|f(x)<\Vert x\Vert^\alpha$ for all $x \in B_r(0)$, prove ...
1
vote
1answer
46 views

Exact expansion of functions

Prove that for any twice differentiable function $f: {R}^n \to R$, $f(y) = f(x) + \nabla f(x)^T (y-x)+ \frac{1}{2} (y-x)^T \nabla^2f(z)(y-x) $, for some $z$ on the line segment $[x, y]$. Note that ...
7
votes
1answer
157 views

taylor's formula in $\mathbb{R}^n$ as exponential of derivative operator

I hope this question is not to vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots $$ and taylor's formula. Is ...
4
votes
2answers
92 views

$f(y) \leq f(x)+\nabla f(x)\cdot (y-x) $ and $f(x)\geq 0$ implies that $f$ is constant.

Here is the question. Suppose that $f: \mathbb R^n \rightarrow \mathbb R$ has two derivatives and the associated hessian matrix is negative semidefinite on all of $\mathbb R^n$. Show that for any ...
0
votes
1answer
118 views

Applying a multidimensional variant of Taylor's Theorem

Given $f:R^n\rightarrow R^n$ continuously differentiable in some convex open set $D$ and $x,x+p\in D$, taylor's theorem is given as: $$f(x+p)=f(x)+\int_0^1 J(x+tp)p \, dt,\text{ where }J\text{ is ...
1
vote
2answers
797 views

Vector taylor series

Classical Electrodynamics by Jackson says "With a Taylor series expansion of the well-behaved $\rho (\mathbf{x'})$ around $\mathbf{x'} = \mathbf{x}$ one finds ..." and then he says basically that ...
2
votes
1answer
70 views

When is the limit in $y$ of a Taylor expansion in $x$ a valid expansion?

I'd be interested to know when, if $$f(x,y)=g(x,y)+O(x^n)$$ we have that $$\lim_{y\rightarrow c}=\lim_{y\rightarrow c}g(x,y)+O(x^n).$$ Are there conditions of $f$ and/or $g$ that make sure that this ...
1
vote
2answers
92 views

Taylor's formula

Taylor's Formula Write taylor's formula for $F(x,y)= \sin(x)\sin(y)$ using $a=0$, $b=0$, and $n=2$. $$\sin(h)\sin(k)=hk−\frac 16h(h^2+3k^2)\cos\theta h\sin\theta k−\frac 16 k(3h^2+k^2)\sin\theta ...
1
vote
1answer
388 views

how do I find the taylor polynomial of multivariable functions?

I know taylor polynomial for single variable functions but I am having trouble understanding how to find taylor polynomials for multivariable functions. I know how to find partial derivatives as well ...
1
vote
1answer
106 views

Product of Taylor polynomials

I'm trying to prove the following proposition: Let $U\in R^n$ be open, and $f,g\colon U\to R$ be $C^k$ functions, then the Taylor polynomial of $fg$ is computed as $P_{f,a}^k(a+\vec{h})\cdot ...
1
vote
1answer
192 views

Taylor series expansion of $\sec(x +y^2)$

We have $f(x,y) = \sec(x+y^2)$ I want to find the first two non-zero terms of $f$ at $(0,0)$ starting by Taking the first few terms of $\cos x$ centered at zero, $1 - \frac{x^2}{2!} $ Using this ...
0
votes
2answers
136 views

Domain of convergence of $f^{-1}: \mathbb R ^N \mapsto \mathbb R^N$ taylor series

In another question, I ask about the topology of the singular manifold of the Jacobian. What i want to ask in here is about the radius of convergence of a Taylor series expansion of the inverse ...
1
vote
1answer
206 views

Taylor expansion in time of the time component of a stress energy tensor

Perform a taylor expansion in 3 dimensions in time on the time compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ given that $r$ is a contstant and $n^{i} y_{i}$ is the scalar product of a ...